Research ArticleMATERIALS SCIENCE

Higher-order topological insulators

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Science Advances  01 Jun 2018:
Vol. 4, no. 6, eaat0346
DOI: 10.1126/sciadv.aat0346
  • Fig. 1 Topologically protected hinge excitations of second-order 3D TIs.

    (A) Time-reversal breaking model with chiral hinge currents running along the corners of a Embedded Image-preserving bulk termination where periodic boundary conditions in z direction are assumed. (B) Time-reversal invariant model with antipropagating Kramers pairs of hinge modes. Highlighted in gray are the planes invariant under the mirror symmetries Embedded Image and Embedded Image that protect the hinge states. (C) By supplementing each surface of the chiral HOTI in (A) with a Chern insulator with Hall conductivity σxy = ± e2/h, the number of chiral hinge modes can be changed by 2. The Hall conductivities of the additional Chern insulator layers alternate (blue for + e2/h, red for − e2/h) to comply with the Embedded Image symmetry. The topology is therefore Embedded Image-classified.

  • Fig. 2 Simple model for a chiral HOTI.

    (A) Schematic phase diagram for model (1), where NI stands for normal insulator. (B) A unit cell of noncollinear magnetic order with Embedded Image symmetry. (C) Energy spectrum of model (1) with chiral hinge currents (red) in the geometry of Fig. 1A. For a slab geometry, where the bulk is terminated in just one direction of space, there are in general no gapless modes.

  • Fig. 3 Bulk-surface-hinge correspondence of helical HOTIs.

    (A) Additional hinge modes obtained by decorating the surfaces with 2D time-reversal symmetric TIs in a mirror-symmetric fashion. They can always be combined in bonding and antibonding pairs {R1 + R2, L1 + L2} and {R1R2, L1L2}, with mirror eigenvalues + i and − i, respectively. Therefore, they do not change the net mirror chirality of the hinge. (B) Mirror-symmetry protected Dirac cones on a (110) surface. (C) Slightly tilting the surface normal out of the mirror plane gaps the Dirac cones and forms a Kramers pair of domain wall states between two surfaces with opposite tilting. The mirror eigenvalues of the hinge modes are tied to those of the Dirac cones, which, in turn, are related to a bulk topological invariant, the mirror Chern number Cm. (D) Further deforming the surface to the (100) and (010) orientation in a mirror symmetry–preserving manner does not change this correspondence.

  • Fig. 4 Helical HOTI emerging from the topological crystalline insulator SnTe.

    (A) Rocksalt lattice structure of SnTe. Uniaxial strain along the (110) direction breaks the mirror symmetries represented by dotted lines but preserves the ones represented by dashed lines. (B) Circles indicate the location of Dirac cones in the surface Brillouin zone of pristine SnTe for various surface terminations. Those crossed by dotted mirror symmetries are gapped in SnTe with uniaxial strain while the others are retained. The two red Dirac cones are enforced by a mirror Chern number Cm = 2, corresponding to one helical pair of hinge modes. k1 is the momentum along the direction with unit vector Embedded Image. (C) DFT band structure of a slab of SnTe with open boundary conditions in the (100) direction under 3% strain in the (110) direction. (D) DFT calculation of the gap Δ that develops on the (100) surface of SnTe under (110) uniaxial strain. (E) DFT-based Wannier tight-binding calculation of SnTe with the (111) ferroelectric displacement in a semi-infinite geometry in which the (Embedded Image) surface and the (Embedded Image) surface meet at a hinge that is parallel to the (111) direction. A single Kramers pair of hinge states is visible. This distortion breaks all mirror symmetries except those with normal (Embedded Image), (Embedded Image), and (Embedded Image), which retain their mirror Chern number 2 for a sufficiently small distortion. The (Embedded Image) and (Embedded Image) surfaces considered here are both not invariant under these mirror symmetries, but the hinge formed between them is invariant under the mirror symmetry with normal (Embedded Image), supporting topological hinge states. (F) Low-energy finite size spectrum of SnTe with uniaxial (110) strain obtained using a tight-binding model (see the Supplementary Materials) for open boundary conditions in the x and y directions (with Lx = Ly = 111 atoms) and periodic boundary conditions in the z direction. States localized in the bulk, on the (100)/(010) surfaces, and on the hinges are color-coded. Near kz = π, four Kramers pairs of hinge modes, one localized on each hinge, are found. Upper left inset: Localization of the gapless modes. Lower left inset: Spatial structure of one such mode near a hinge. Only a small portion of the lattice near the hinge is shown. Right inset: Electronic structure of undistorted SnTe in the same geometry, showing two “flat band” hinge modes in addition to the gapless surface Dirac cones. (G) Topological coaxial cable geometry to realize (110) uniaxial displacement. A Si or SiO substrate (gray) is etched to have a rhombohedral cross section and then coated with SnTe (blue) yielding Kramers pairs of hinge modes (orange).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/6/eaat0346/DC1

    Supplementary Text

    fig. S1. Nested entanglement and Wilson loop spectra for the second-order 3D chiral TI model defined in Eq. 1 in the main text with M/t = 2 and Δ1/t = Δ2/t = 1.

    fig. S2. Real-space hopping picture for the optical lattice model with Formula.

    fig. S3. Real-space structure for a chiral HOTI.

    fig. S4. Constraints on mirror-symmetric domain wall modes in two dimensions.

    fig. S5. Wilson loop characterization of helical HOTIs.

    fig. S6. Band structure of the surface Dirac cones of the topological crystalline insulator SnTe calculated in a slab geometry.

    fig. S7. High-symmetry points in the BZ of SnTe.

    References (3640)

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Text
    • fig. S1. Nested entanglement and Wilson loop spectra for the second-order 3D
      chiral TI model defined in Eq. 1 in the main text with M/t = 2 and Δ1/t = Δ2/t = 1.
    • fig. S2. Real-space hopping picture for the optical lattice model H4 = Σij›e(x,y)(tij(x,y)cz,icz,j + tijzcz,icz+1, j) .
    • fig. S3. Real-space structure for a chiral HOTI.
    • fig. S4. Constraints on mirror-symmetric domain wall modes in two dimensions.
    • fig. S5. Wilson loop characterization of helical HOTIs.
    • fig. S6. Band structure of the surface Dirac cones of the topological crystalline insulator SnTe calculated in a slab geometry.
    • fig. S7. High-symmetry points in the BZ of SnTe.
    • References (36–40)

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